arXiv subjects
Publications and source records attributed to Jakub Pachocki.
We show that schemes for sparsifying matrices based on iteratively resampling rows yield guarantees matching classic 'offline' sparsifiers (see e.g. Spielman and Srivastava [STOC 2008]). In particular, this gives a formal analysis of a scheme very similar to the one proposed by Kelner and Levin [TCS 2013].
Finding a small spectral approximation for a tall $n \times d$ matrix $A$ is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of $A$. Row sampling improves interpretability, saves space when $A$ is sparse, and preserves row structure, which is especially important, for example, when $A$ represents a graph. However, correctly sampling rows from $A$ can be costly when the matrix is large and cannot be stored and processed in memory. Hence, a number of recent publications focus on row sampling in the streaming setting, using little more space than what is required to store the outputted approximation [KL13, KLM+14]. Inspired by a growing body of work on online algorithms for machine learning and data analysis, we extend this work to a more restrictive online setting: we read rows of $A$ one by one and immediately decide whether each row should be kept in the spectral approximation or discarded, without ever retracting these decisions. We present an extremely simple algorithm that approximates $A$ up to multiplicative error $ε$ and additive error $δ$ using $O(d \log d \log(ε||A||_2/δ)/ε^2)$ online samples, with memory overhead proportional to the cost of storing the spectral approximation. We also present an algorithm that uses $O(d^2$) memory but only requires $O(d\log(ε||A||_2/δ)/ε^2)$ samples, which we show is optimal. Our methods are clean and intuitive, allow for lower memory usage than prior work, and expose new theoretical properties of leverage score based matrix approximation.
We introduce the notion of balance for directed graphs: a weighted directed graph is $α$-balanced if for every cut $S \subseteq V$, the total weight of edges going from $S$ to $V\setminus S$ is within factor $α$ of the total weight of edges going from $V\setminus S$ to $S$. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with $α= 1$) and residual graphs of $(1+ε)$-approximate undirected maximum flows (with $α=O(1/ε)$). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably harder in directed graphs. We first revisit oblivious routings in directed graphs. Our main algorithmic result is an oblivious routing scheme for single-source instances that achieve an $O(α\cdot \log^3 n / \log \log n)$ competitive ratio. In the process, we make several technical contributions which may be of independent interest. In particular, we give an efficient algorithm for computing low-radius decompositions of directed graphs parameterized by balance. We also define and construct low-stretch arborescences, a generalization of low-stretch spanning trees to directed graphs. On the negative side, we present new lower bounds for oblivious routing problems on directed graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs is $Ω(n)$ in general; this result improves upon the long-standing best known lower bound of $Ω(\sqrt{n})$ given by Hajiaghayi, Kleinberg, Leighton and Räcke in 2006. We also show that our restriction to single-source instances is necessary by showing an $Ω(\sqrt{n})$ lower bound for multiple-source oblivious routing in Eulerian graphs. We also give a fast algorithm for the maximum flow problem in balanced directed graphs.
We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of $|V(H)|^{o(|V(H)|)}$-time algorithm deciding if graph $G$ is a subgraph of $H$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems. Moreover, as a consequence of our reductions conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.
Subgraph Isomorphism is a very basic graph problem, where given two graphs $G$ and $H$ one is to check whether $G$ is a subgraph of $H$. Despite its simple definition, the Subgraph Isomorphism problem turns out to be very broad, as it generalizes problems such as Clique, $r$-Coloring, Hamiltonicity, Set Packing and Bandwidth. However, for all of the mentioned problems $2^{\mathcal{O}(n)}$ time algorithms exist, so a natural and frequently asked question in the past was whether there exists a $2^{\mathcal{O}(n)}$ time algorithm for Subgraph Isomorphism. In the monograph of Fomin and Kratsch [Springer'10] this question is highlighted as an open problem, among few others. Our main result is a reduction from 3-SAT, producing a subexponential number of sublinear instances of the Subgraph Isomorphism problem. In particular, our reduction implies a $2^{Ω(n \sqrt{\log n})}$ lower bound for Subgraph Isomorphism under the Exponential Time Hypothesis. This shows that there exist classes of graphs that are strictly harder to embed than cliques or Hamiltonian cycles. The core of our reduction consists of two steps. First, we preprocess and pack variables and clauses of a 3-SAT formula into groups of logarithmic size. However, the grouping is not arbitrary, since as a result we obtain only a limited interaction between the groups. In the second step, we overcome the technical hardness of encoding evaluations as permutations by a simple, yet fruitful scheme of guessing the sizes of preimages of an arbitrary mapping, reducing the case of arbitrary mapping to bijections. In fact, when applying this step to a recent independent result of Fomin et al.[arXiv:1502.05447 (2015)], who showed hardness of Graph Homomorphism, we can transfer their hardness result to Subgraph Isomorphism, implying a nearly tight lower bound of $2^{Ω(n \log n / \log \log n)}$.
We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all `long' Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems.